We consider a fixed circle (C) tangent to a fixed line Δ at a given point O of this line.

Circles Γ tangent to circles C in M and to the right Δ in N are studied.

Show that the MN line passes through a fixed point I. Infer that the circles Γ remain orthogonal to a fixed circle.

My code is :

restart; with(geometry);with(plots);

_EnvHorizontalName := 'x';_EnvVerticalName := 'y';

dist := proc (M, N) sqrt(Vdot(expand(M-N), expand(M-N))) end proc;

point(oo, 0, 3); p := 6;

point(N, 5, 0);

line(Delta, y = 0, [x, y]);

para := x^2 = 2*p*y;

solve(subs(x = 5, para), y); point(varpi, 5, 25/12);

line(alpha, [oo, varpi]); k := 3/(25/12);

point(M, (0+5*k)/(1+k), (3+25*k*(1/12))/(1+k));

circle(C, x^2+(y-3)^2 = 9, [x, y]);cir := implicitplot(x^2+(y-3)^2 = 9, x = -5 .. 5, y = -5 .. 7, color = blue);

Para := implicitplot(para, x = -40 .. 40, y = 0 .. 40, linestyle = 3, color = coral);

homothety(J, N, -k, M); coordinates(J);

circle(C1, (y-25/12)^2+(x-5)^2 = (25/12)^2, [x, y]);line(lNJ, [N, J]);

triangle(T1, [J, oo, M]); triangle(T2, [N, varpi, M]);

C1 := implicitplot((y-25/12)^2+(x-5)^2 = (25/12)^2, x = 2 .. 8, y = 0 .. 5, color = magenta);dr1 := draw([oo, Delta, varpi, N, M, J], printtext = true); dr2 := draw([alpha(color = black), lNJ(color = black), T1(color = green, filled = true), T2(color = green, filled = true)]);

inversion(M, M, C);

inversion(N, M, C);

Fig := proc (xOm)

local cir, c2, C2, C1, c3, C3, k, M, N, J, sol, dr, varpi;

global p, para, Para;

sol := solve(subs(x = xOm, para), y);

cir := (y-sol)^2+(x-xOm)^2 = sol^2; c2 := x^2+(y-3)^2 = 9;

geometry:-point(N, xOm, 0); sol := solve(subs(x = xOm, para), y);

geometry:-point(varpi, xOm, sol); k := 3/sol;

geometry:-point(M, xOm*k/(k+1), (3+k*sol)/(k+1));

geometry:-homothety(J, N, -k, M);

c3 := (x-(1/2)*xOm)^2+(y-3)^2 = (1/4)*dist(N, J)^2;

C1 := plots:-implicitplot(cir, x = -xOm .. 3*xOm, y = 0 .. 3*xOm, color = magenta);

C2 := plots:-implicitplot(c2, x = -xOm .. 2*xOm, y = 0 .. 2*xOm, color = blue);

C3 := plots:-implicitplot(c3, x = -xOm .. 2*xOm, y = 0 .. 2*xOm, color = blue);

dr := geometry:-draw([varpi, M, J]);

plots:-display([Para, C2, C1, C3, dr], view = [-xOm .. 3*xOm, -1 .. 3*xOm], axes = normal, scaling = constrained) end proc;

Fig(8);

display([seq(Fig(4+.8*i), i = 4 .. 15)]);

display({C1, Para, cir, dr1, dr2}, view = [-8 .. 8, -1 .. 8], axes = normal, scaling = constrained, size = [500, 500]);

I don't know what is that orthogonal circle to each tangent circles. Thank you to help me.

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